Integral: An Easy Approach After Kurzweil and HenstockCambridge University Press, 20 d’abr. 2000 - 311 pàgines Integration has a long history: its roots can be traced as far back as the ancient Greeks. The first genuinely rigorous definition of an integral was that given by Riemann, and further (more general, and so more useful) definitions have since been given by Lebesgue, Denjoy, Perron, Kurzweil and Henstock, and this culminated in the work of McShane. This textbook provides an introduction to this theory, and it presents a unified yet elementary approach that is suitable for beginning graduate and final year undergraduate students. |
Continguts
Introduction | 1 |
12 Notation and the Riemann definition | 3 |
13 Basic theorems upper and lower integrals | 6 |
14 Differentiability continuity and integrability | 10 |
15 Limit and Rintegration | 16 |
16 Exercises | 18 |
Basic Theory | 22 |
23 Cousins lemma | 25 |
The SLintegral | 151 |
42 SLintegration | 155 |
43 Limit and SLintegration | 162 |
44 Equivalence with the KHintegral | 169 |
Generalized AC Functions | 175 |
52 Uniformly AC functions | 177 |
53 AC and VB on a set | 179 |
54 ACG functions | 187 |
231 Applications of Cousins lemma | 26 |
24 The definition | 29 |
25 Basic theorems | 32 |
26 The Fundamental Theorem of calculus | 46 |
27 Consequences of the Fundamental Theorem | 50 |
28 Improper integrals | 55 |
29 Integrals over unbounded intervals | 59 |
210 Alternative approach to integration over unbounded intervals | 64 |
211 Negligible sets | 66 |
212 Complex valued function | 69 |
213 Exercises | 71 |
Development of the Theory | 76 |
32 Henstocks lemma | 81 |
33 Functions of bounded variation | 83 |
34 Absolute integrability | 86 |
35 Limit and KHintegration | 88 |
36 Absolute continuity | 100 |
37 Equiintegrability | 104 |
371 The second mean value theorem | 108 |
38 Differentiation of integrals | 110 |
39 Characterization of the KHintegral | 112 |
310 Lebesgue points approximation by step functions | 115 |
311 Measurable functions and sets | 117 |
3111 A nonmeasurable set | 126 |
312 The McShane integral | 127 |
3121 A short proof | 135 |
3131 F Riesz definition | 139 |
3132 Quick proofs | 140 |
315 Exercises | 145 |
55 Controlled convergence | 190 |
56 Exercise | 197 |
Integration in Several Dimensions | 202 |
611 Sets in ℝ | 203 |
62 Divisions partitions | 204 |
63 The definition | 210 |
64 Basic theorems | 215 |
641 Prelude to Fubinis theorem | 217 |
65 Other theorems in ℝ | 220 |
653 Absolute integrability | 221 |
654 Convergence measurability AC | 223 |
66 The Fubini theorem | 229 |
67 Change of variables | 236 |
672 Notation lemmas | 240 |
673 The theorem | 241 |
68 Exercises | 248 |
7 Some Applications | 252 |
72 A line integral | 253 |
721 Greens theorem | 267 |
722 The Cauchy theorem | 274 |
73 Differentiation of series | 279 |
74 Dirichlets problem and the Poisson integral | 282 |
75 Summability of Fourier series | 286 |
76 Fourier series and the space 𝓛² | 289 |
77 Exercises | 295 |
Appendix 1 Supplements | 299 |
| 305 | |
| 308 | |
Frases i termes més freqüents
8-fine partition absolutely continuous absolutely integrable absolutely KH-integrable bounded interval bounded variation Cauchy characteristic function closed interval Consequently continuous function COROLLARY countable Cousin's lemma d-fine defined definition denote derivative differentiable disjoint dominated convergence theorem equation equiintegrable everywhere example EXERCISE exists f is absolutely F is AC F is continuous f is integrable f is KH-integrable finite fn(x follows Fourier series Fubini theorem function f Fundamental Theorem gauge Henstock's lemma Hint holds inequality integrable functions integral of ƒ KH-integrable function Lebesgue integral Let f limit M-integrable McShane integral measure zero monotone convergence theorem non-negative open intervals open set positive ɛ primitive proof of Theorem prove Riemann integrable Riemann sum right-hand side satisfied Section semicontinuous sequence f set of measure SL-integrable step functions tagged partial division theory triangle π π